Understanding the definition of isometric manifolds

Question

I would like to understand the definition of isometric spacetimes in this article on page 3. I do not understand the sentence highlighted. How do we define $\varphi_*$ in plain mathematical language?

Answer 1

The concepts used are actually two:

1. Pullback: given a smooth map between two manifolds $M,N$, the pullback is a linear map: $$\varphi^*:T_sN\to T_sM\\ a\mapsto \varphi^*(a)(x,v_1,\dots,v_s)=a(\varphi(x),\text{d}\varphi(v_1),\dots,\text{d}\varphi(v_n))$$

The pullback is always defined for covariant tensor fields. That's enough to define an isomorphism: two Riemannian manifolds $(M,g_M);(N,g_N)$ are isomorphic if $\varphi$ is a diffeomorphism between the two of them and $\varphi^*(g_N)=g_M$

2. Pushforward: the idea is the same of point (1). The pushforward, however, is not uniquely defined (or at least not elementary) unless the map $\varphi$ is invertible. In such a case, the pushforward is defined as $$\varphi_*:T^kM\to T^kM\\ b^{i_1\dotsi_k}(x)e_{i_1}\otimes\dots\otimes e_{i_k}\mapsto \varphi_*(b)= b^{i_1\dotsi_k}(\varphi^{-1}(y))\text{d}\varphi(e_{i_1})\otimes\dots\otimes \text{d}\varphi(e_{i_k})$$

Given a diffeomorphism, we can combine these two tools to "move" any tensor field from $M$ to $N$: the contravariant part is moved with the pushforward $\varphi_*$ the covariant part with $(\varphi^{-1})^*$